Integrality of Tate-cycles

Abstract

We explain a technical result about p-adic cohomology and apply it to the study of Shimura varieties. The technical result applies to algebraic varieties with torsion-free cohomology, but for simplicity we only treat abelian varieties.

Suppose A is an abelian variety over V, a p-adic discrete valuation ring with perfect residue field k. Let V₀ = W(k) ⊆ V denote the maximal unramified subring, V₀ ⊆ K₀ and V ⊆ K the fraction fields. If π is a uniformizer of V, then π satisfies an Eisenstein equation f(π) = 0, and V ≅ V₀[T]/(f(T)). Let R_V denote the p-adically completed PD-hull of V₀[T] along (f(T)).

Associated to A there are the étale cohomology

1

and the crystalline cohomology

2

The étale cohomology H_étⁱ(A) is a free Z_p-module with a continuous action of Gal(/K), while H_crⁱ(A) is a filtered free R_V-module with a Frobenius-endomorphism Φ. These are related by Fontaine’s isomorphism

3

which after inverting p allows one to recover one cohomology from the other.

An étale Tate cycle of degree r is a Galois-invariant element

4

A crystalline Tate cycle of degree r is an element

5

which lies in the r − th stage of the Hodge filtration and is annihilated by Φ − p^r.

By Fontaine’s comparison the ℚ_p-vector spaces of étale and crystalline Tate cycles are isomorphic. We show:

Theorem.

If r ≤ p − 2 then ψ_ét is integral, if and only if, the corresponding ψ_cr is integral.

The proof uses techniques developed previously.

A. Vasiu (2) has used this result to show that certain Shimura varieties classifying abelian varieties with higher-order Tate cycles have good reduction. He obtains smooth models for them by normalizing the moduli-space of abelian varieties in the generic fiber of the Shimura variety. To control this normalization one uses the valuative criterion, together with the theorem applied to the Tate cycles defining the Shimura variety.